This is a problem for a introducing Linear Algebra Mathematics class.
The complete question states: I want to pick a collection of vectors in the plane $\mathbb{R}^2$ so that each pair has a strictly negative dot product like so: $v_i*v_j < 0$ for any pair of i and j where $i \ne j$.
How many vectors can I have in my collection?
Give an example if a "biggest collection" of such vectors
How do you know you cannot choose any more than that?
I know that the dot products between to vectors is always negative when $\frac{\pi}{2}\lt \theta <= \pi$. What I don't understand is how to apply this fact to answer the questions.
Hint: Assume you have such a collection $(v_i)$. By replacing each $v_i$ with $\frac{v_i}{\| v_i \|}$, we won't change the sign of the dot products so we can assume $\| v_i \| = 1$ for all $i$. So how many vectors can you choose on the unit circle when the angle between each pair of vectors is greater than $\frac{\pi}{2}$ but less than $\frac{3\pi}{2}$?